**2. Problem Formulation**

In the current research, it is assumed that a 2D unsteady magneto-forced convective flow of ferrofluid past a radiate stretchable surface with impacts on Navier slip and convective heating are additionally considered. In this work, Cobalt is considered and is treated as a base nanoparticle, with kerosene as a base ferrofluid. The stretchable surface switches on from a fine slot, which is positioned at the starting point of a 2D coordinate system (*<sup>x</sup>*, *y*). At this point, the x-axis is considered all along the stretching direction of the sheet, having stretched velocity *Uw* = *ax*, which is applied vertically to the sheet externally. A constant magnetic strength *B*0 is applied normal to the sheet. The mathematical model describing the system is (see Chamkha [29])

$$
\frac{\partial \mu}{\partial \mathbf{x}} + \frac{\partial v}{\partial y} = 0 \tag{1}
$$

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \frac{\mu\_{ff}}{\rho\_{ff}} \frac{\partial^2 u}{\partial y^2} - \frac{\sigma\_{ff} B\_0^2}{\rho\_{ff}} u \tag{2}$$

$$\alpha \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \alpha\_{ff} \left( \frac{\partial^2 T}{\partial y^2} - \frac{1}{k\_{ff}} \frac{\partial q^r}{\partial y} \right) \tag{3}$$

Subjected to the corresponding boundary conditions (see [30–34]):

$$\begin{cases} u(t, \mathbf{x}, 0) = \mathcal{U}\_{\overline{w}} + \mathcal{L}\mu\_{ff} \frac{\partial u}{\partial y}, v(t, \mathbf{x}, 0) = 0, -k\_{ff} \frac{\partial T}{\partial y}(t, \mathbf{x}, 0) = h\_f(T\_f - T) \\ u(t, \mathbf{x}, \boldsymbol{\omega}) = 0, T(t, \mathbf{x}, \boldsymbol{\omega}) = T\_{\boldsymbol{\omega}}. \end{cases} \tag{4}$$

where *t*, *u* and *v* are the time and velocity components along the *x* and *y* axes and *T* is the temperature in the fluid phase. ρ*f f* stands for the density. μ*f f* stands for viscosity. β*f f* Stands for the ferrofluid volumetric thermal expansion coefficient. <sup>σ</sup>ff stands for electrical conductivity. <sup>α</sup>*f f* = *k f f* /(ρ*Cp*)*f f* stands for the thermal diffusivity of the ferrofluid. *L* stands for the slip coefficient, which represents Navier slip, and *hf* stands for the heat transfer coefficient. *Tf* stands for the uniform temperature of the stretchable surface. *k*ff stands for the thermal conductivity of ferrofluid. (ρ*Cp*)*f f* stands for the specific heat of the ferrofluid at a constant pressure. The radiative heat flux qr is approached according to the Rosseland approximation (see [35,36]):

$$\frac{\partial \eta^{r}}{\partial y} = -\frac{4\sigma\_{1}}{3\beta\_{R}} \frac{\partial T^{4}}{\partial y} \tag{5}$$

where β*R* and σ1 stand for the mean absorption coefficient and the Stefan–Boltzmann constant. As carried out by Raptis [35], the fluid-phase temperature variations within the flow are approached to be adequately tiny so that *T*<sup>4</sup> may be obvious as a linear function of temperature. This is created by extending *T*<sup>4</sup> in a Taylor series on the free-stream temperature *T*∞ and removing higher-order terms to yield

$$T^4 = 4T^4\_{\infty}T - 3T^4\_{\infty} \tag{6}$$

By applying Equations (5) and (6) in the last term of Equation (3), we obtain

$$\frac{\partial q^{r}}{\partial y} = -\frac{16\sigma\_{1}T\_{\infty}^{3}}{3\beta\_{R}}\frac{\partial^{2}T}{\partial y^{2}}\tag{7}$$

In the current investigation, the following thermophysical relations are applied [37];

$$\begin{array}{lcl}\rho\_{ff} & (1-\chi)\rho\_f + \chi\rho\_{s\cdot}\ \mu\_{ff} & = \frac{\mu\_f}{(1-\chi)^{2\cdot 5}}, \; a\_{ff} = \frac{k\_{ff}}{\left(\rho\mathbb{C}\_p\right)\_{ff}},\\ \left(\rho\mathbb{C}\_p\right)\_{ff} & = (1-\chi)\left(\rho\mathbb{C}\_p\right)\_f + \chi\left(\rho\mathbb{C}\_p\right)\_{f'}\left(\rho\mathbb{R}\right)\_{ff} & = (1-\chi)\left(\rho\mathbb{R}\right)\_f + \chi\left(\rho\mathbb{R}\right)\_{s'}\\ \frac{k\_{ff}}{k\_f} & = \frac{\left(k\_s+2k\_f\right)-2\chi\left(k\_f-k\_s\right)}{\left(k\_s+2k\_f\right)+\chi\left(k\_f-k\_s\right)}, \; \frac{\sigma\_{ff}}{\sigma\_f} & = 1+\frac{3(\chi-1)\_X}{(\chi+2)-(\chi-1)\_X}\text{where }\chi &=\frac{a\_p}{\sigma\_f}\end{array} \tag{8}$$

Here, χ is nanoparticle volume fraction. Table 1 represents the thermophysical properties of ferrofluid.

**Table 1.** Thermophysical properties of kerosene, water and cobalt [37].


In this stage, the expressions for *u*, *v*, and θ will be defined as:

$$u = \frac{\partial \Psi}{\partial y}, v = -\frac{\partial \Psi}{\partial \mathbf{x}}, \theta = \frac{(T - T\_{\infty})}{(T\_f - T\_{\infty})'} \tag{9}$$

Substituting Equations (7)–(9) into Equations (1)–(4), we obtain

$$\frac{\partial^2 \Psi}{\partial t \partial y} + \frac{\partial \Psi}{\partial y} \frac{\partial^2 \Psi}{\partial x \partial y} - \frac{\partial \Psi}{\partial x} \frac{\partial^2 \Psi}{\partial y^2} = \nu\_f \Xi\_1 \frac{\partial^3 \Psi}{\partial y^3} - \frac{\sigma\_{ff}}{\sigma\_f} \frac{B\_0^2 \sigma\_f}{\rho\_f} \frac{1}{1 - \phi + \phi(\rho\_s/\rho\_f)} \frac{\partial \Psi}{\partial y} \tag{10}$$

$$\frac{\partial\theta}{\partial t} + \frac{\partial\Psi}{\partial y}\frac{\partial\theta}{\partial x} - \frac{\partial\Psi}{\partial x}\frac{\partial\theta}{\partial y} = \frac{\upsilon\_f}{\text{Pr}}\Xi\_2 \Big(\frac{k\_{ff}}{k\_f} + \frac{4}{3}\text{R}d\Big)\frac{\partial^2\theta}{\partial y^2} \tag{11}$$

$$\begin{array}{ll}\frac{\partial\mathbf{w}}{\partial y} & = \; \mathbf{a}\mathbf{x} + \mathbf{L}\mu\_{ff}\frac{\partial^2\mathbf{w}}{\partial y^2}, \frac{\partial\mathbf{w}}{\partial\mathbf{x}} & = \; 0, \frac{k\_{ff}}{k\_f}\frac{\partial\mathbf{0}}{\partial y} = -\frac{h\_f}{k\_f}(1-\theta)\; \mathbf{a}\; \mathbf{y} & = \; \mathbf{0}, \\\\ \frac{\partial\mathbf{w}}{\partial\mathbf{y}} & = \; 0, \theta & = \; \mathbf{0}, \mathbf{a}\mathbf{t}\; \mathbf{y} \to \infty. \end{array} \tag{12}$$

$$\text{where } \Xi\_1 = \frac{1}{\left(1 - \chi\right)^{2.5} \left[1 - \chi + \chi(\rho\_s/\rho\_f)\right]}, \Xi\_2 = \frac{1}{\left[1 - \chi + \chi(\rho \mathbb{C}\_p)\_s/\left(\rho \mathbb{C}\_p\right)\_f\right]}$$
